Inversion of some irrational Laplace transforms

A general method for the inversion of Laplace transforms which are rational functions in the variablep ^1/2 is presented. The numerical aspects of the method are discussed.     

Performance of numerical inversion of Laplace transforms

The Laplace transformation is one of the most powerful tools in mathematics. However, the analytical inversion of a function in the Laplace field to the real field may sometimes be very difficult, or impossible. In 1970, Stehfest presented an algorithm to invert numerically these Laplace-field functions. The purpose of this work is to discuss the performance of this algorithm, and present a Turbo Pascal 5.0 program to perform numerical inversions of Laplace-field functions.     

Linear system identification using numerical computation of Laplace transforms

A method for system identification using sampled values of the initial transient step or impulse response is described. A polynomial fit of the sampled values is made using Lagrange interpolation and the Laplace transform of the output observed is determined. Then the coefficients of the numerator and denominator polynomials of the system transfer function are determined by minimizing the square of the difference between the observed and calculated values of the Laplace transform of the output variable at a number of discrete points. This process is considerably simplified by the use of tables of coefficients for the numerical calculation of Laplace transforms.     

Functional approximation for inversion of Laplace transforms via polynomial series

A method for finding the inverse of Laplace transforms using polynomial series is discussed. It is known that any polynomial series basis vector can be transformed into Taylor polynomials by use of a suitable transformation. In this paper, the cross product of a polynomial series basis vector is derived in terms of Taylor polynomials, and as a result the inverse of the Laplace transform is obtained, using the most commonly used polynomial series such as Legendre, Chebyshev, and Laguerre. Properties of Taylor series are first briefly presented and the required function is given as a Taylor series with unknown coefficients. Each Laplace transform is converted into a set of simultaneous linear algebraic equations that can be solved to evaluate Taylor series coefficients. The inverse Laplace transform using other polynomial series is then obtained by transforming the properties of the Taylor series to other polynomial series. The method is simple and convenient for digital computation. Illustrative examples are also given,     

Numerical inversion of multidimensional Laplace transforms by the Laguerre method

Numerical transform inversion can be useful to solve stochastic models arising in the performance evaluation of telecommunications and computer systems. We contribute to this technique in this paper by extending our recently developed variant of the Laguerre method for numerically inverting Laplace transforms to multidimensional Laplace transforms. An important application of multidimensional inversion is to calculate time-dependent performance measures of stochastic systems. Key features of our new algorithm are: (1) an efficient FFT-based extension of our previously developed variant of the Fourierseries method to calculate the coefficients of the multidimensional Laguerre generating function, and (2) systematic methods for scaling to accelerate convergence of infinite series, using Wynn's ε-algorithm and exploiting geometric decay rates of Laguerre coefficients. These features greatly speed up the algorithm while controlling errors. We illustrate the effectiveness of our algorithm through numerical examples. For many problems, hundreds of function evaluations can be computed in just a few seconds.     

Solution of piecewise-linear ordinary differential equations using waveform relaxation and Laplace transforms

Piecewise-linear (PWL) functions are frequently used to describe the nonlinear branch equations of nonlinear devices in LSI circuits. New techniques for the solution of the differential equations describing the behavior of piecewise-linear circuits will be presented. These techniques are based on the waveform relaxation method to decouple the system equations and Laplace transform techniques to solve the decoupled equations. Several desirable features of the resulting algorithm are discussed.     

On a functional approximation for inversion of Laplace transforms via shifted Chebyshev series

A functional representation for inversion of the Laplace transform of a function is considered. The function is given as a shifted Chebyshev series expansion. Using special operational properties, each Laplace transform is converted into a set of simultaneous linear algebraic equations that are then easily solved to give the coefficients of the Chebyshev series. The method is simple and very suitable for computer programming. Applications to rational and irrational Laplace transforms are presented to demonstrate the satisfactory results that the method provides.     

Fuzzy transforms for compression and decompression of color videos

We use fuzzy transforms for coding/decoding color frames of videos and we compare the results with the same frames reconstructed with the standard JPEG method. We classify all the frames in intra-frames and predictive frames by adopting a similarity measure based on the Lukasiewicz t-norm and a pre-processing phase which determines the best similarity threshold value. The compression is made on particular frames, called Δ-frames, obtained from a suitable difference defined on the values of the pixels of an intra-frame and a predictive frame. Under high compression rates, we see that the Peak Signal to Noise Ratio of the frames obtained with the fuzzy transforms is averagely close to the PSNR obtained with the JPEG method. We use the videos at URL sampl.eng.ohiostate.edu/∼sampl/database.htm.     

Fuzzy transforms method and attribute dependency in data analysis

In this paper, we present a method based on fuzzy transforms to establish dependencies between numerical attributes in datasets. We find the best fuzzy partitions of the attribute domains with respect to which we determine the direct and inverse fuzzy transforms. We use two specific regression indexes (which must be smaller than a threshold deduced experimentally) for evaluating dependency between numerical attributes. The experiments are conducted on two well known datasets: “El Nino” (http://kdd.ics.uci.edu/databases/el_nino/el_nino.data.html) and the remote sensing data determined from US Forest Service (Region 2, Resource Information System, http://kdd.ics.uci.edu/databases/covertype/covertype.data.html). Our results are quite in agreement with the regression analysis of the same data.     

Fuzzy transforms of monotone functions with application to image compression

This paper is focused on the special properties of functions (monotonicity, Lipschitz continuity) that are invariant with fuzzy transform. On the basis of the monotonicity invariance, we propose an efficient algorithm for improved image compression and reconstruction based on fuzzy transform.     

Hierarchical generalized Hough transforms and line-segment based generalized Hough transforms

Generalized Hough transforms are fast point pattern matching algorithms which have applications in image processing. Two extensions to generalized Hough transforms are described. The first allows for matching hierarchically organized point patterns. An example concerning image registration using hierarchical point patterns is described. The second extension concerns matching patterns of simple geometric objects, such as line segments, rather than patterns of points.     

Multidimensional logical transforms

This correspondence reviews the logical transform and pre-sents new theoretical and empirical developments. Novel extensions of the logical transform into spaces of three and four dimensions are de-scribed using the hexahedral and octahedral tessellations. These new tessellations are important in that they yield sufficiently compact neighborhoods to permit multidimensional cellular logic transforms to be carried out in real time by table lookup. Applications of logical transforms in these tessellations to data and image analysis are provided.     

一种基于拉普拉斯方程的图像彩色化方法

黑白图像的彩色化是图像处理领域一个活跃的、有挑战性的研究课题。在总结现有图像彩色化方法的基础上，分析了彩色化由局部向全局扩展的本质，进而提出了一种通过求解拉普拉斯方程实现颜色扩展的彩色化方法。为提高边界定位能力，设计了加权形式的差分方案。与同是采用偏微分方程的Saprio彩色化方法相比，该方法与其控制颜色扩展的机制不同，图像彩色化效果要明显优于Saprio方法。     

Mellin transforms and asymptotics

Mellin transforms and Dirichlet series are useful in quantifying periodicity phenomena present in recursive divide-and-conquer algorithms. This note illustrates the techniques by providing a precise analysis of the standard topdown recursive mergesort algorithm, in the average case, as well as in the worst and best cases. It also derives the variance and shows that the cost of mergesort has a Gaussian limiting distribution. The approach is applicable to a number of divide-and-conquer recurrences.